A new wavefront sensing scheme derived from curvature wavefront sensing is presented. Non-linear Curvature wavefront sensing (NLCWFS) is a scheme derived from conventional curvature wavefront sensing. NLCWFS uses four defocused pupil images and a non-linear wavefront reconstruction. NLCWFS is largely achromatic and does not require a flat (<1 rad RMS) wavefront to operate: it is therefore very robust, and is an attractive alternative to less efficient Shack-Hartmann wavefront sensing. For low order aberrations, sensitivity gains corresponding to 5 and 8 stellar magnitudes can be achieved over the conventional Shack-Hartmann sensor on respectively a 8-mtelescope and a 30-m telescope.